In general relativity, a spacetime is said to be static if it does not change over time and is also irrotational. It is a special case of a stationary spacetime, which is the geometry of a stationary spacetime that does not change in time but can rotate. Thus, the Kerr solution provides an example of a stationary spacetime that is not static; the non-rotating Schwarzschild solution is an example that is static.

Formally, a spacetime is static if it admits a global, non-vanishing, timelike Killing vector field K {\displaystyle K} which is irrotational, i.e., whose orthogonal distribution is involutive. (Note that the leaves of the associated foliation are necessarily space-like hypersurfaces.) Thus, a static spacetime is a stationary spacetime satisfying this additional integrability condition. These spacetimes form one of the simplest classes of Lorentzian manifolds.

Locally, every static spacetime looks like a standard static spacetime which is a Lorentzian warped product R × {\displaystyle \times } S with a metric of the form

g [ ( t , x ) ] = − β ( x ) d t 2 + g S [ x ] {\displaystyle g[(t,x)]=-\beta (x)dt^{2}+g_{S}[x]} ,

where R is the real line, g S {\displaystyle g_{S}} is a (positive definite) metric and β {\displaystyle \beta } is a positive function on the Riemannian manifold S.

In such a local coordinate representation the Killing field K {\displaystyle K} may be identified with ∂ t {\displaystyle \partial _{t}} and S, the manifold of K {\displaystyle K} -trajectories, may be regarded as the instantaneous 3-space of stationary observers. If λ {\displaystyle \lambda } is the square of the norm of the Killing vector field, λ = g ( K , K ) {\displaystyle \lambda =g(K,K)} , both λ {\displaystyle \lambda } and g S {\displaystyle g_{S}} are independent of time (in fact λ = − β ( x ) {\displaystyle \lambda =-\beta (x)} ). It is from the latter fact that a static spacetime obtains its name, as the geometry of the space-like slice S does not change over time.